Lorentzian Wormhole Topologies

• Lorentzian wormhole topologies are smooth, time-oriented spacetimes that connect distinct regions via a minimal throat satisfying flare-out conditions.
• They utilize methods such as topological surgery and controlled energy condition violations to enable topology change and maintain causal traversability.
• These geometries impact gravitational path integrals, holography, and stability analyses, offering insights into quantum corrections and factorization puzzles.

A Lorentzian wormhole is a smooth, time-oriented manifold with Lorentzian metric that admits nontrivial causally traversable connections between separated spatial regions or asymptotic infinities. Unlike Euclidean instantons, Lorentzian wormhole topologies embed directly into physical spacetime, permitting analysis of causality, energy condition violations, and topology change in real time. The construction, classification, and physical implications of such geometries have become central in high-energy theory, quantum gravity, and holography, with significant advances illuminating their stability, physical realization, and role in the gravitational path integral.

1. Wormhole Topology and Geometric Structures

A Lorentzian wormhole is characterized by a spacetime topology connecting two (or more) asymptotic regions, with a minimal area spatial submanifold—the throat—satisfying flare-out conditions. In classical four-dimensional General Relativity (GR), the prototypical example is the Morris–Thorne wormhole, whose spatial slices are $\mathbb{R} \times S^{2}$ and are described by

$$ds^2 = - e^{2\Phi(r)} dt^2 + \dfrac{dr^2}{1-\frac{b(r)}{r}} + r^2 d\Omega^2_{2}$$

where $b(r)$ satisfies $b(r_0) = r_0$ at the throat and the flare-out condition $b'(r_0) < 1$ determines traversability (Cataldo et al., 2011, Cruz et al., 16 Jan 2024). For higher-dimensional and generalized settings, spatial slices are of type $\mathbb{R} \times S^{N-1}$ (Cataldo et al., 2011), or can acquire further structure via extra dimensions, nontrivial fiberings, or warped products (Kar, 2022).

Key topological features include:

• Handle attachment: Time evolution of a slice through the wormhole corresponds to a cobordism where a handle is attached, interpolating between topologically distinct hypersurfaces (Pisana et al., 4 May 2025).
• Connected sum: For topology change, a 0-surgery replaces two 3-balls by a handle $D^{1} \times S^{2}$, yielding a global topology change while preserving smooth structure except possibly at isolated points or regions (Pisana et al., 4 May 2025).
• Topology with extra dimensions: In higher-dimensional models, the topology is often $\mathbb{R} \times S^{D-3} \times S^{1}$, with the $S^{1}$ factor shrinking to zero at the throat ("degenerate throat") (Kar, 2022).
• Nonstandard traversability: In certain string-theoretic models, the manifold has regions ("non-Riemannian" loci) where only extended objects such as chiral strings traverse, and point-particle geodesics cannot (Jang et al., 5 Dec 2024).

2. Construction and Realization Methods

Classical GR and Energy Conditions: In $N+1$ Einstein gravity with or without cosmological constant, wormholes require matter violating the Null Energy Condition (NEC) near the throat, typically realized with phantom or barotropic fluids ($p_r = \omega_r \rho$) (Cataldo et al., 2011, Cruz et al., 16 Jan 2024). In higher dimensions, warping extra dimensions enables solutions in vacuum or with normal matter (Kar, 2022).

Topological Surgery and Morse Theory: Topology change in Lorentzian geometry is forbidden except under singular conditions (Geroch’s theorem). A rigorous Lorentzian wormhole nucleation via 0-surgery (handle attachment) produces a critical locus, which is made nonsingular by connected sum with a closed 4-manifold such as $\mathbb{CP}^2$ (Pisana et al., 4 May 2025). This excises the would-be singularity and replaces it with a compact region harboring closed timelike curves (CTCs).

Quantum Gravity and Path Integrals: In axion gravity or Einstein–3-form systems, wormhole contributions arise as complex or real saddles of the Lorentzian path integral, classified via Picard–Lefschetz theory (Loges et al., 2022). Giddings-Strominger (GS) wormholes survive as quadratic-action minima, with their stability and dominance depending on boundary conditions and bilocal operator insertions.

Stringy Constructions: Nontrivial NS–NS $H$-flux and dilaton profiles support "wine-glass" wormholes, where the global geometry is smooth in Double Field Theory but contains chiral accessibility loci for strings (Jang et al., 5 Dec 2024). These backgrounds evade standard energy condition constraints due to the negative kinetic dilaton.

3. Topology Change, Causality, and Chronology Violation

Crotches and Singularities: Topology change in Lorentzian signature inevitably introduces codimension-2 singular loci ("crotches"), at which the metric determinant vanishes (Blommaert et al., 2023). These crotches can be regulated so that away from the handle-attachment, the spacetime remains smooth. Each crotch realizes a local "pair-of-pants" (handle addition), and in 2D the Euler characteristic drops by one.

Avoidance of Singularities: The connected sum with $\overline{\mathbb{CP}}^2$ replaces crotch singularities by smooth chronology-violating regions containing CTCs confined inside a compact set (Pisana et al., 4 May 2025). The Morse region stays stably causal, and the only causality violation is in the $\overline{\mathbb{CP}}^2$ pocket.

Gauge Equivalence with Euclidean Wormholes: The inclusion of Lorentzian crotches can be shown (in models such as JT gravity) to be gauge-equivalent to the inclusion of smooth, mostly Euclidean metrics with handles, linking the Lorentzian and Euclidean path integral approaches (Blommaert et al., 2023).

4. Classification and Parameter Dependence

Lorentzian wormhole topologies can be classified via the nature of the saddle points in the gravitational action or path integral:

• Real Lorentzian saddles: these traverse two turning points, with no Euclidean region; classic time-symmetric wormholes (Loges et al., 2022).
• Complex saddles: interpolating between Lorentzian and Euclidean evolution, with partial tunneling behavior.
• Purely Euclidean saddles: e.g., GS wormhole segments, contributing topology-changing but exponentially suppressed amplitudes.
• Higher-genus extensions: Multiple crotches produce genus-$g$ wormholes, and summing over genus reproduces semiclassical expansions of real-time observables (Blommaert et al., 2023). The moduli space of crotch locations correspond to collective coordinates in these expansions.

Physical and geometric features vary by the model:

• Dimension ($N$): Higher dimensions allow vacuum wormhole solutions with warped extra dimensions (Kar, 2022), while lower dimensions admit richer identification possibilities (Cataldo et al., 2011).
• Matter content: NEC-violating fluids, exotic matter, or quantum corrections (LQG parameters $\mathcal{P}, a_0$) control throat size, flare-out, and degree of energy condition violation (Cruz et al., 16 Jan 2024).
• String/particle distinctions: Topologies may impose nontrivial restrictions: in (Jang et al., 5 Dec 2024), only chiral strings traverse the full manifold, while point particles are locally confined.
• Parameter space: Quantum parameters in LQG models ($\mathcal{P}, a_0$) control the throat "smearing," soften curvature, and attenuate exotic matter requirements (Cruz et al., 16 Jan 2024). In dynamical collapse models, specific parameter choices guarantee that the radius of the throat never shrinks to zero, avoiding singularities (Chakrabarti et al., 2021).

5. Physical Implications, Stability, and Observability

Energy Condition Violation: Lorentzian wormholes typically violate the NEC and frequently the WEC, SEC, and DEC near the throat or in chronology-violating cores (Cataldo et al., 2011, Pisana et al., 4 May 2025, Cruz et al., 16 Jan 2024). In models with topological surgery or chiral string traversability, these violations are geometric and unavoidable, in line with singularity and topology-change theorems (Pisana et al., 4 May 2025).

Stability: The Euclidean GS wormhole (axion–gravity) serves as a perturbatively stable saddle once boundary conditions are properly chosen in the 3-form frame—no negative modes appear in the spectrum of fluctuations around the saddle (Loges et al., 2022). Traversable wormholes built from quantum-corrected gravity (LQG) are everywhere regular and maintain the throat for appropriate parameter ranges, while energy condition violations are mitigated but not eliminated (Cruz et al., 16 Jan 2024).

Observational Signatures: Effective potentials for geodesics and congruence properties (expansion, caustics) can in principle discriminate higher-dimensional or exotic wormhole topologies. For warped extra-dimensional wormholes, the presence of an extra-dimensional momentum blocks access to the throat or shifts the caustic location in congruence expansion (Kar, 2022).

Causal Structure: In collapse-induced scenarios, the global topology evolves from trivial to wormhole, but the throat persists (with nonzero areal radius) in the asymptotic future, evading singularity formation (a "no-horizon" or black-bounce phase) (Chakrabarti et al., 2021). In topological-surgery constructions, CTCs are confined and do not cause global causality failure (Pisana et al., 4 May 2025).

6. Role in Quantum Gravity, Holography, and Path Integrals

Lorentzian wormhole topologies have direct implications for the gravitational path integral and quantum gravity:

• Path integral saddle structure: Picard–Lefschetz theory specifies which saddle points (and thus which wormhole topologies) contribute for given boundary data. The inclusion of bilocal operators modifies the contour of integration, adding new Lorentzian saddles (Loges et al., 2022).
• Spectral statistics and genus expansion: In AdS/JT gravity, genus-$g$ Lorentzian wormholes built from crotches reproduce the ramp and plateau regimes in the spectral form factor, matching results from random matrix theory and Euclidean replica wormholes. This substantiates conjectures linking real-time and Euclidean semiclassical sum-over-geometries (Blommaert et al., 2023).
• Factorization puzzles: In AdS/CFT, the existence of (even only Euclidean) wormholes leads to apparent nonfactorization of boundary correlators. The survival and stability of Lorentzian wormhole topologies in the gravitational path integral exacerbates these puzzles (Loges et al., 2022).
• Quantum corrections: Loop quantum gravity and double field theory/DFT examples demonstrate how quantum structure or non-Riemannian points can stabilize or regularize the wormhole throat and reduce the exotic matter requirement to negligibility (Cruz et al., 16 Jan 2024, Jang et al., 5 Dec 2024).

7. Overview Table: Representative Wormhole Topologies

Model/Reference	Topological Feature	Supporting Physics
Classical Morris–Thorne (Cataldo et al., 2011)	$\mathbb{R} \times S^{N-1}$; two asymptotic regions	Phantom/barotropic fluid, NEC-violation
Warped Extra Dimensions (Kar, 2022)	$\mathbb{R} \times S^{D-3} \times S^1$; degenerate S^1 at throat	D≥5 vacuum or normal matter
Topological Surgery (Pisana et al., 4 May 2025)	Connected sum near Morse critical point; compact CTC pocket	Classical GR, 0-surgery, $\overline{\mathbb{CP}}^2$ glue-in
AdS crotches (Blommaert et al., 2023)	Codimension-2 crotches, genus-g wormholes	Delta-function curvature, instanton action on extremal surfaces
Axion/3-form path integral (Loges et al., 2022)	Selection of Lorentzian/complex/Euclidean saddles	Picard–Lefschetz theory, axion or 3-form action
LQG-inspired (Cruz et al., 16 Jan 2024)	Regular throat, two asymptotic regions	Quantum geometry, parameter-tuned exoticity reduction
String DFT (Jang et al., 5 Dec 2024)	$\mathbb{R}^2 \times S^2$ w/ two non-Riemannian S^2 loci	NS–NS H-flux, negative kinetic dilaton; string-chiral traversability

• (Cataldo et al., 2011) Cataldo et al., "N-dimensional static and evolving Lorentzian wormholes with cosmological constant"
• (Loges et al., 2022) Loges, Shiu, Sudhir, "Complex Saddles and Euclidean Wormholes in the Lorentzian Path Integral"
• (Blommaert et al., 2023) Engelhardt, Marolf, Maxfield, "The power of Lorentzian wormholes"
• (Pisana et al., 4 May 2025) Dappiaggi, Surya, "Wormhole Nucleation via Topological Surgery in Lorentzian Geometry"
• (Kar, 2022) Kar, "Wormholes with a warped extra dimension?"
• (Jang et al., 5 Dec 2024) Hohm, Sakatani, "Traversable wormhole for string, but not for particle"
• (Cruz et al., 16 Jan 2024) Lobo, "Traversable Wormholes from Loop Quantum Gravity"
• (Chakrabarti et al., 2021) Chakrabarti, Kar, "A wormhole geometry from gravitational collapse"

Lorentzian wormhole topologies are thus a multifaceted subject at the interface of differential topology, Lorentzian geometry, quantum field theory, and string theory. Their precise realization, physical viability, and implications for quantum gravity remain an active area of high technical and conceptual sophistication.